TOPOLOGY OPTIMIZATION AND OPTIMAL CONTROL IN STRUCTURAL DESIGN

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1 Prodings of COBEM st Brzilin Congrss of Mhnil Enginring Otobr 24-28, 211, Ntl, RN, Brzil OPOLOGY OPIMIZAION AND OPIMAL CONROL IN SRUCURAL DESIGN Alxndr Moltr, Vldir Bottg, Univrsidd Fdrl d Plots, Dp. Mthmtis nd Sttistis, Cmpus Univrsitário, s/nº, 354, 961-9, Plots, RS, Brzil. Jun S. O. Fons, jun@ufrgs.br Otávio A. Alvs d Silvir, otvio.silvir@ufrgs.br Univrsidd Fdrl do Rio Grnd do Sul, Dp. Mhnil Enginring, R. Srmnto Lit, 425, 95-17, Porto Algr, RS Brzil. Abstrt. his work prsnt struturl dsign mthodology onsidring ontrol ffts, th hng of th topology by ontrol for tion, nd dsign modl ontrol for suitbl fixd tutor lotions. h tutors r omposd by pizoltri mtril. h topology optimiztion in this work uss homogniztion dsign mthod, bsd on th onpt of optimizing th mtril distribution, through dnsity distribution, whil th ontrol for is obtind by th optiml ontrol dsign for trnsint rspons nd prformd in th modl sp. A Continuum finit lmnts modling is pplid to simult th dynmi hrtristis of th strutur. h ost funtionl is th strin nrgy of th strutur nd th ontrol nrgy. Rsults of numril simultions for ntilvr bm modl r prsntd nd disussd. Kywords: opologi Optimiztion, tutors, dynmis, vibrtion ontrol. 1. INRODUCION Struturl vibrtion ontrol is prtiulrly importnt onsidrtion in th dsign of dynmi systms. h min id of th struturl optimiztion is to obtin n optiml mtril lyout of lod-bring strutur. Usully, ontinuum topology optimiztion problms r formultd to minimiz th struturl mtril volum or to optimiz th struturl prformn. A typil xmpl is to ris th first fundmntl frquny of strutur whil obying volum onstrint (Zhn, Xioming nd Rui, 29). Mnwhil, struturl dynmis ontrol, onsidring pizoltri tutors, is usd to minimiz supprss vibrtion ffts. h dsign of ths struturs tks into ount th intrtion of th pplid fors, th lsti mods nd th tutor plmnt. hr r lwys fundmntl intrst in dsigns with ffiint struturl ontrol systm from both struturl nd ontrol nginrs. Howvr, ths groups hv bn working indpndntly. rditionlly, th struturl dsignr dvlops his dsign bsd on strngth nd stiffnss rquirmnts, nd th ontrol dsignr rts th ontrol lgorithm to rdu th dynmi rspons of strutur (Ou nd Kikuhi, 1996). In this work w r dsigning th strutur nd ontrols simultnously, mning tht th ost funtion inluds not only th strin nrgy, but lso th ontrol nrgy. h rson why topology optimiztion is boming vry importnt rsrh fild is th nssity of ffiint mthodologis to dsign struturs, thus sving mtril nd tim. h min objtiv of th topology optimiztion problm is to find mtril distribution tht minimizs givn objtiv funtionl, subjtd to st of onstrints, hivd by onsistnt prmtriztion of th mtril proprtis in h prt of th dsign domin. A nturl qustion is whthr thr xists or not mtril in givn point, whih lds to disrt problm. It is wll-known tht this intgr prmtriztion lds to numril diffiultis, ssoitd with th intgr problm onvrgn (Crdoso nd Fons, 23; Bndsø nd Kikuhi, 1988; Bndsø nd Sigmund, 1999). Minimizing th vibrtion ffts of th dynmi rspons is n importnt gol for th struturl vibrtion ontrol, nd th fftivity of th ontrol dpnds on th wighting mtris. h objtiv of this ppr is to prsnt struturl dsign mthodology onsidring th ontrol ffts, th hng of th topology by ontrol for tion, nd dsign modl ontrol for suitbl fixd tutor lotions. h struturl optimiztion dsign is ompltd through dnsity dsign mthod, whil th ontrol for is obtind by th optiml ontrol dsign for trnsint rspons nd prformd in th modl sp. h ffiint struturl ontrol dsign nds rful sltion of tutor positions (Ou nd Kikuhi, 1996). Howvr, in this work th tutors lotions r hosn rbitrrily prior to th struturl dsign. In ft, it is wll known tht good lotion for n tutor in ntilvr strutur is los th fixd siz of th strutur, sin it ts upon th first nd most signifint mod (Sun t l., 24, Donoso, A. nd Sigmund, O., 29; Moltr, A. t l, 21). h lowr fundmntl mods r rsponsibl for th most of th tip displmnt of th bm; thrfor, th first two ignfuntions r omputd nd onsidrd in this work. h dynmis nd ontrol dsign wr inludd in topology optimiztion od. Simultions wr ondutd to ssss th fftivnss nd ontrol modl ffiiny.

2 Prodings of COBEM st Brzilin Congrss of Mhnil Enginring Otobr 24-28, 211, Ntl, RN, Brzil 2. FORMULAION OF SRUCURAL OPOLOGY OPIMIZAION CONSIDE-RING CONROL ACION h homogniztion dsign mthod (Bndsø nd Sigmund, 23) is hoosing th tool for th topology optimiztion onsidring ontrol tion. his mthod is bsd on th onpt of optimizing th mtril distribution, through dnsity distribution. A finit lmnt msh is dfind to prform th struturl modl nlysis (Bth, 1996). As simplifition, w ssum tht th dnsity is onstnt in h finit lmnt. An optimlity ritri (OC) is drivd from th nssry minimiztion onditions, nd it is solvd to updt th dnsity distribution. A numbr of simplifitions r introdud to th implmnttion, s rgulr msh. W now onsidr tht th objtiv funtion is th sum of th strin nrgy nd th ontrol nrgy. hn, th topologi optimiztion problm in stdy stt hs th form min J, J ( x) = f Rf + U QU x subjt to V ( x) V min < x x 1 min = V KU = Hf + F (1) whr U nx1 is th displmnt vtor, H nxm is lotion mtrix for th ontrol for, m is th numbr of tion ontrol fors nd F nx1 is th pplid xtrnl for vtor, f mx1 is n pplid ontrol for in trms of ltri lod. h mgnituds of th mtris Q nxn nd R mxm r ssignd ording to th rltiv importn of th stt vribls nd th ontrol for in th minimiztion produr. K nxn is th finit lmnt globl stiffnss mtrix. x is th vtor of dsign vribls, x min is vtor of minimum rltiv dnsitis. V(x) nd V is th mtril volum nd dsign domin volum, rsptivly nd V min is th prsribd volum funtion. Considring th disrtiztion, N = ( x ) p U QU u q u, (2) = 1 whr N is th numbr of lmnts, p is th pnliztion xponnt, u nd q r th lmnt displmnt vtor nd wighting mtrix, rsptivly. h optimiztion problm is solvd using th Optimlity ritrion (OC), nd this ritrion is drivd from th Krush-Kuhn-ukr onditions (Bndsø nd Kikuhi, 1988). h Lgrngin funtion of th minimiztion problm is N N ( ) min 1 2 min 3 mx = 1 = 1 L( x) = J ( x) + λ ( V x V V ) + λ ( KU ( Hf + F)) + λ ( x x ) + λ ( x x ). (3) whr th slrλ nd th vtor λ1r th globl Lgrngin multiplirs, nd th slrs multiplirs for lowr nd uppr sid onstrints. o lot sttionry point, it is nssry tht L / x =, thn λ nd λ r Lgrngin 2 3 L J V ( KU ( Hf + F)) = + λ + λ λ + λ = x x x x (4) Hr, w ssum tht onstrins of th dsign vribls r not tiv, λ = λ = nd tht th lod nd fors r 2 3 ( Hf + F) dsign indpndnt, =. x h fdbk rquirs full knowldg of stts. By using th displmnt losd-loop fdbk ontrol w n ssum 1 f = R H U, (5) thn th quilibrium onstrint from Eq. (1) boms K U = F, (6)

3 Prodings of COBEM st Brzilin Congrss of Mhnil Enginring Otobr 24-28, 211, Ntl, RN, Brzil whr 1 K = K + HR H. (7) W n not tht this K is th modifid mtrix undr ontrol fft nd th modifition pprs whr th for ontrol is pplid, whih ffts lso th ignvlus nd displmnt of th strutur. h problm n b solvd s th onvntionl stti finit lmnt mthod in stndrd form K U = F. h influn of th wighting mtrix R is n importnt spt to onsidr. o hv signifint fft on th topology of th strutur, th mtrix R -1 nd n quivlnt mgnitud omptibl with th stiffnss mtrix. Sin th stiffnss is modifid on h itrtion, thn R is hosn s R = Gdig ( P / µ ), whr µ r th ignvlus (th smllst to th lrgst), G is wighting onstnt is nd P mxm is omputd s th nrgti quivlnt gnrlizd for (Yng t l., 25; Kumr nd Nrynn, 28; Moltr t l, 21), tht is, s momnt Snsitivity Anlysis Snsitivitis r dfind s th drivtivs of th objtiv funtion nd th onstrints with rspt to th dsign vribls, nd is oftn th mjor omputtionl ost of th optimiztion. In this modl, th objtiv funtion snsitivity rquirs diffrntiting displmnts (whih implis stiffnss diffrntition) nd ignvlus. Substituting Eq. (5) in J from Eq. (1), this yild ( ) J = U ΓU + U QU = U Γ + Q U, (8) whr Γ = HR RRH. h drivtivs of J n b omputing by J U U = ( Γ + Q) U + U( Γ + Q) x x x. (9) It is possibl obtin simplifition for th drivtivs of J, Eq (9), djusting th mtrix Q by Q=K. hn, th objtiv funtion hng, substituting Eq. (7) into Eq. (6) nd thn Eq. (6) into Eq. (1), this yild ( ) J = f Rf + U Hf + F. (1) Using Eq. (5) into Eq. (1), w obtin th simplifid objtiv funtion J = F U. (11) king th drivtiv of th objtiv funtion, on h lmnt, on n obtin J U = F x x, (12) nd substituting nd U x = K K x 1 U (13) K K 1 µ = + H H x x GP x (14) into (12), w hv

4 Prodings of COBEM st Brzilin Congrss of Mhnil Enginring Otobr 24-28, 211, Ntl, RN, Brzil J K 1 µ = U U + U H H U. (15) x x GP x h snsitivity of th h ignvlu µ is omputd by (Hftk t l., 199) ( K - µ M) φ = (16) nd µ K M k m = φ µ φ = φ µ φ, (17) x x x x x whr φ is th mss-normlizd ignvtor nd M is th mss mtrix, on h lmnt φ, k nd m. With som xpnding of th trms, lso simplifition of th qutions nd huristis shm for th dsign vribls (Sigmund, 21), w n obtin th nw x for h itrtion: ( k ) ( k ) α ( k ) ( xmin x δ ) x B ( xmin x δ ) α α ( min δ ) ( δ ) ( k ) ( k ) ( k ) α ( x + δ ) ( x + δ ) x B mx, if mx,, ( k + 1) ( k ) ( k ) ( k ) ( k ) x = x B if mx x, x x B min 1, x +, min 1, if min 1,, (18) whr δ nd α r rsptivly, th prsribd mov limit nd th prsribd numril dmping offiint. h trm B is dfind from th optimlity ondition s B = ( ) p x p 1 u k u, (19) λρ V whr ρ is th mss dnsity of th mtril, V is th lmntl volum. h msh-indpndnt filtr is providd from Sigmund (21). 3. CONROL EFFECS ON SRUCURAL OPOLOGY W n imgin priori tht ontrol fors ting in diffrnt lotions on th strutur should influn th optimizd dsign. o xmplify this ft, w onsidr dsign domin s ntilvr bm shows in Fig Figur 1. Dsign domin ntilvr bm. For struturl only dsign of this domin, w us th omplin s th objtiv funtion, nd obtin th topology, for th ntilvr bm, shown in Fig. 2. hn w try to introdu ontrol fors on this dsign lyout. It is possibl tht on th dsird lotion for th tutor thr is no mtril. If th optimiztion is prformd without onsidring th ontrol fors, thn w nd ithr to hng th tutor lotion or to rdsign th strutur. In Fig. 2 w inditd with points (smll irls) th tutor lotion nd dsignd th strutur gin, this tim onsidring th ontrol for. h vlu of th prmtrs r: p 3, α.5, δ.2 P = dig -.2, -.1 nd G is djustd with = = =, ( ) similr mgnitud of th stiffnss invrs vlus. h nw topology for this problm is shown in Fig. 2b, orrsponding to th dsign domin, Fig. 1. h msh domin in th simultion uss 144 finit lmnts.

5 Prodings of COBEM st Brzilin Congrss of Mhnil Enginring Otobr 24-28, 211, Ntl, RN, Brzil (2) (2b) Figur 2. -opology optimiztion without ontrol for tion. b-opology optimiztion with ontrol for tion. In this simultions it n b notd tht th strutur dsign hng ompltly with th ontrol tion ffts. Additionlly som ttntion for th tutor lotion is rquird to ssur th ontrollbility of th systm nd th vlu of G nd th tutor lotion hs onsidrbl influn of ontrol ffiiny nd th finl topology. 4. OPIMAL CONROL DESIGN IN MODAL SPACE Aftr omputing th optiml strutur w srh for th vibrtion supprssion for trnsint rspons of th systm. It is possibl to dsign th ontrol for th displmnt of prtiulr point of th strutur. In this work w driv th ontrol in indpndnt modl sp, omputing th bhvior of th systm on th nods whr th ontrol fors r pplid. h formultion of indpndnt modl sp ontrol, drivd by th lssil optiml thory (Nidu, 23), ssoitd with th distributd-prmtr systm n b writtn brifly s follows. h modl formultion for th systm is ɺɺ, (2) 2 η + ω η = φ GP V whr ω r th frqunis nd V th voltg pplid to tutor. Lt ssum tht l = (m + numbr of mods). hn, Glxl is wighting mtrix nd P l x l is th nrgti quivlnt gnrlizd for mtrix. h dynmi systm dfind by Eq. (2) n b prmtrizd in first ordr qutions nd writtn in th stt-offiint form ɺ (21) y = Ay + BV ;, z = Cy whr y 2lx1 is stt, tim dpndnt vribl, sp, yɺ 2l x1 is th vtor of th first ordr tim drivts of th stts in modl l V S R is th ontrol vtor, S is th ontrol onstrint st. z 2lx1 is snsor output nd C 2lx2l is th snsor output mtrix in modl sp. his systm rprsnts th onstrins from th nonlinr rgultor problm, togthr with y( t ) = y, y( ) =, rsptivly th initil nd finl onditions. h offiint mtris, in modl sp, without onsidring dmping, r givn by l xl Ιl xl l xl η l xl C l xl A =,,, 2 B = = l xl l xl φ y C = ω GP ηɺ l xl l xl (22) A stt fdbk rthr thn output fdbk is doptd to nhn th ontrol prformn. h qudrti ost funtion for th rgultor problm is givn by J 1 = + 2 t y Qy V RV dt, (23) whr Q is smi-positiv-dfinit wighting mtrix on th outputs nd R positiv dfinit wighting mtrix on th ontrol inputs. Assuming full stt fdbk, th ontrol lw is givn by

6 Prodings of COBEM st Brzilin Congrss of Mhnil Enginring Otobr 24-28, 211, Ntl, RN, Brzil V R B Pz Kz, (24) 1 = = whr P stisfis th lgbri Riti qution 1 A P + PA PBR B P + Q =. (25) h omputtionl ost is high if ll mods r onsidrd. But it n b drmtilly rdud if only fw mods r dominnt nd thir ontrol is suffiint for th whol strutur. h losd loop dynmis of th systm is givn by ɺ, (26) y = ( ) A - BKC y h stbility of th fdbk mtrix A = ( ) ontrol. It n b show tht for our problm th stbility for A is ssurd. A - BKC is n importnt ondition for th xistn of th fdbk 5. RESULS h physil systm onsidrd in th simultions is omposd by ntilvrd stl bm shown in Fig. 2 nd th pizoltri tutor bondd on th uppr surf, t th bginning of th bm. h snsor is onsidrd pizofilm bondd on th bottom surf, lso t th bginning of th bm. h rsulting topology for this problm is show in Fig. 3 nd Fig. 3b, whr th lotions of th horizontl ontrol fors r indit by points (smll irls) nd th snsor by smll rtngl. his lotion for th tutor ws hosn bus it is known ft tht th bst pl for on tutor, bondd on ntilvr bm, is s los s possibl to th fixd siz of th strutur, whih brs th mximum strss indud by th first nd most signifint mod. Som simplifitions r introdud to th problm nd its rspons nlysis. W ssum tht th two ontrol points n hv diffrnt fors. his mns tht thr r two xtrnl tutors. Only on tutor would gnrt qul mgnitud opposing fors nd nd to b xpliitly inludd in th modl. () (b) Figur 3. -opology optimiztion without ontrol for tion. b-opology optimiztion with ontrol for tion. It n b not in Fig. 3b tht th topology hs not hngd s muh s in Fig. 2b. his n b ttributd to lotion nd mgnitud of th for ontrol hv bn lss inisiv in th topology thn in th prvious ss, but with nhnd ontrol ffiiny. h onvrgn of th objtiv funtion is plottd in Fig. 4. J J J Itrtion Numbr Itrtion Numbr Itrtion Numbr () (b) ()

7 Prodings of COBEM st Brzilin Congrss of Mhnil Enginring Otobr 24-28, 211, Ntl, RN, Brzil Figur 4. Objtiv funtion onvrgn of th ntilvr bm s - without ontrol; b- with ontrol for t th nd of th bm ; - with ontrol fors t th bginning of th bm. W n obsrv in Fig. 4 tht th onvrgn is fstr in th initil 3 itrtions, ftr thr is smllr hng of th objtiv funtion vlu t h itrtion. h topology shown in Figur 2 subjt to trnsint fors produs initil dformtion nd so tiv th nturl vibrtions. h thr fr vibrtion mods of th modl, whih finit lmnt disrtiztion r shown in Fig. 5, whos frqunis r 5Hz, 5Hz nd.24hz, rsptivly. mod1 mod2 mod3 Figur 5. Dfltions of th mods in modl sp. In Fig. 5, it n b not tht th first mod is not vrtil nd th sond nd third mods r vrtil mods. h rsults of th optiml ontrol simultion in Mtlb r shown in Fig. 6. h wighting mtris nd ontrol G = dig(1,1,1,1 ), Q = dig(1,1,1,1) mtris r:. Hr r onsidrd th two first mods of th optimizd strutur. h position 1 is on th lft point (smll irl) nd position 2 on th right, shown in Fig. 3. h fourth-ordr Rung-Kutt mthod ws usd to intgrt th qutions for thirty sonds simultion. displmnt mod 1, position 1 displmnt mod 2, position φ 1 φ t displmnt mod 1, position t displmnt mod 2, position φ 1 φ t t Figur 6. Dfltions of first nd sond mods without indpndnt modl ontrol (blu nd rd) nd with indpndnt modl ontrol (blk). It is possibl obsrv tht th modl displmnt go quikly to zro, vn without nturl dmping. h hoi of th bst vlus for th stt nd ontrol wighting mtris Q nd R is importnt. A good hoi n improv th ffiiny of th ontrollrs. In this ppr w hv tstd som wighting mtris nd onludd tht, for our ontrol dsign, th good rsults r obtind round th vlus hooss bov. Smllr or grtr vlus ffts th ontrol ffiiny.

8 Prodings of COBEM st Brzilin Congrss of Mhnil Enginring Otobr 24-28, 211, Ntl, RN, Brzil 6. ACKNOWLEDGEMENS h uthor A. Moltr knowldg th finnil support of FAPERGS (Fundção d Ampro à Psquis do Estdo do Rio Grnd do Sul ) Porto Algr - Rio Grnd do Sul -Brzil, Projt: 1/ REFERENCES Bth, J. K. nd Wilson, E. L., 1976, Numril Mthods in Finit Elmnt Anlysis, Prnti Hll.. Bndsø, M. P. nd Kikuhi, N., Gnrting Optiml opologis in Struturl Dsign Using Homogniztion Mthod. Comp. Mth. Appl. Mh. Eng., Vol. 71, pp Bndsø, M. P. nd Sigmund, O., Mtril Intrpoltion Shms in opology Optimiztion. Arhiv of Applid Mhnis, Vol. 69, pp Bndsø, M. P. nd Sigmund, O., 23, opology Optimiztion hory, Mthods nd Applitions, Springr, Nw York. Crdoso, E. L. nd Fons, J., S., O., 23. Complxity Control in th opology Optimiztion of Continuum Struturs. Journl of th Brz. So. of Mh. Si. & Eng., Vol. 3, pp Donoso, A. nd Sigmund, O., 29. Optimiztion of pizoltri bimorph tutors with tiv dmping for stti nd dynmi lods. Struturl nd Multidisiplinry Optimiztion, Vol. 38, No. 4, pp Hftk, R.., Gurdl, Z. nd Kmt, M. P., 199, Elmnts of Struturl Optimiztion, Kluwr. Kumr, K. R., Nrynn, S., 28. Ativ vibrtion ontrol of bms with optiml plmnt of pizoltri snsor/tutor pirs. Smrt Mtrils nd Struturs, Vol. 17, pp Moltr, A., Silvir, O. A. A., Fons, J. nd Bottg, V., 21. Simultnous Pizoltri tutor nd Snsor Plmnt Optimiztion nd Control Dsign of Mnipultors with Flxibl Links Using SDRE Mthod. Mthmtil problms in Enginring, Vol. 21, pp Nidu, D. S., 23, Optiml Control Systms, CRC Prss, Nw York. Ou, J.-S. nd Kikuhi, N., Intgrtd optiml struturl nd vibrtion ontrol dsign. Struturl Optimiztion, Vol. 12, pp Sigmund O., 21, A 99 lin topology optimiztion od writtn in Mtlb. Struturl Multidisiplinry Optimiztion, Vol. 21, pp Yng, Y., Jin, Z., Soh, C.K., 25. Intgrtd optiml dsign of vibrtion ontrol systm for smrt bms using gnti lgorithms. Journl of Sound nd Vibrtion, Vol. 282, pp Zhn K., Xioming W. nd Rui W., 29. opology optimiztion of sp vhil struturs onsidring ttitud ontrol ffort. Finit Elmnts in Anlysis nd Dsign, Vol. 45, pp RESPONSIBILIY NOICE h uthors Alxndr Moltr, Vldir Bottg Jun S. O. Fons nd Otávio A. Alvs d Silvir r th only rsponsibl for th printd mtril inludd in this ppr.